5 min read•december 29, 2022

Peter Cao

Jed Quiaoit

Building on the stuff introduced in the previous section, there are many ways in which we can represent data from two categorical variables. Some of these are more graphical, like **side-by-side bar graphs**, **segmented bar graphs**, and **mosaic plots**, while others are numerical, like **two-way tables** (also called contingency tables). 🏟️

As you can see below, two-way tables show how the individuals are distributed in the cells relative to other variables.

Here is an example of a two-way table:

This table succinctly summarized a survey of 4826 people where the participants were asked about their chance of getting rich. We can see that 194 of the participants felt that they had almost no chance of getting rich and that 2367 of the participants were females. We can also see that 758 males thought that there was a good chance that they could get rich.

**Joint Relative Frequencies**

We can also put relative frequencies in a two-way table, where the frequencies in the table are a proportion out of the whole survey sample size. A joint relative frequency is one that has the proportion that an individual surveyed shares two characteristics, for example, being female and almost certain. The joint relative frequency for this is 486/4826 as there are 486 males who are almost certain and 4826 total people were sampled. In a joint relative frequency two-way table, the overall total in the bottom right will always be 1.00. 1️⃣

Here is an example of a side-to-side bar graph:

Here, we have the dividing category being gender and each of the bars is the percentage of each gender who have a certain opinion about age and wealth. 💰

In this case, we can see that the cumulative frequency for each gender is 100%. This allows us to compare the size of the different responses given by one category.

Another way to think of a mosaic plot is that it is a graphic version of a two-way table. Mosaic plots can be useful for showing the relationship between two categorical variables and for comparing the proportions of data points in different categories. Here is an example of a mosaic plot:

In each of these, we can show if these categorical variables are associated or not. If they are associated, then we can say that they are also correlated or dependent. To do this, compare the heights (or widths) of the corresponding segments across different categories, of the heights are significantly different, then we can say that they are dependent, as this means that a certain group in one category may be more likely to have a certain response in another category but it doesn’t mean that there is a cause and effect relationship between the two. 🪢

For example, if we have data on the class level (junior, senior, etc.) and whether or not students in each class finish their homework on time, we can create a side-by-side bar graph or mosaic plot to compare the proportions of students in each class who finish their homework on time. If the proportions are significantly different across the classes, it suggests that there is an *association* between the two variables.

When analyzing bivariate categorical data, we can also use various methods to determine if the two variables are associated or not. If the variables are associated, it means that there is a pattern or trend in the data that suggests a relationship between the variables.

In general, it's important to keep in mind that finding an association between two variables does not necessarily mean that there is a cause-and-effect relationship between the variables. **Correlation does not imply (or equal) causation**. There may be other factors that are influencing the relationship between the two variables, and it is important to consider these factors when interpreting the data! 🎀

🎥 **Watch: AP Stats - ****Exploring Two Variable Data**

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👆Unit 1 – Exploring One-Variable Data

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📊Unit 5 – Sampling Distributions

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😼Unit 7 – Means

✳️Unit 8 – Chi-Squares

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